THE ASYMPTOTIC DISTRIBUTION OF THE LENGTH OF BETA-COALESCENT TREES

• 成果类型: Article
• 署名作者: Kersting, Goetz
• 署名单位: Goethe University Frankfurt
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP827
• 发表日期: 2012
• 页码: 2086-2107
• 关键词: segregating sites; number

摘要

We derive the asymptotic distribution of the total length L-n of a Beta(2 - alpha, alpha)-coalescent tree for 1 < alpha < 2, starting from n individuals. There are two regimes: If alpha 1/2(1 + root 5), then L-n suitably resealed has a stable limit distribution of index alpha. Otherwise L-n just has to be shifted by a constant (depending on n) to get convergence to a nondegenerate limit distribution. As a consequence, we obtain the limit distribution of the number S-n of segregation sites. These are points (mutations), which are placed on the tree's branches according to a Poisson point process with constant rate.